The global dimension of the algebras of polynomial integro-differential operators 𝕀n and the Jacobian algebras 𝔸n

The aim of the paper is to prove two conjectures from the paper [V. V. Bavula, The algebra of integro-differential operators on a polynomial algebra, J. London Math. Soc. (2) 83 (2011) 517–543, arXiv:math.RA/0912.0723] that the (left and right) global dimension of the algebra [Formula: see text] of polynomial integro-differential operators and the Jacobian algebra [Formula: see text] is equal to [Formula: see text] (over a field of characteristic zero). The algebras [Formula: see text] and [Formula: see text] are neither left nor right Noetherian and [Formula: see text]. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. An analogue of Hilbert’s Syzygy Theorem is proven for the algebras [Formula: see text], [Formula: see text] and their factor algebras. It is proven that the global dimension of all prime factor algebras of the algebras [Formula: see text] and [Formula: see text] is [Formula: see text] and the weak global dimension of all the factor algebras of [Formula: see text] and [Formula: see text] is [Formula: see text].

CATEGORICAL TINKERTOYS FOR ${\mathcal N} = 2$ GAUGE THEORIES

In view of classification of the quiver 4d [Formula: see text] supersymmetric gauge theories, we discuss the characterization of the quivers with superpotential [Formula: see text] associated to a [Formula: see text] QFT which, in some corner of its parameter space, looks like a gauge theory with gauge group G. The basic idea is that the Abelian category [Formula: see text] of (finite-dimensional) representations of the Jacobian algebra [Formula: see text] should enjoy what we call the Ringel property of type G; in particular, [Formula: see text] should contain a universal "generic" subcategory, which depends only on the gauge group G, capturing the universality of the gauge sector. More precisely, there is a family of "light" subcategories [Formula: see text], indexed by points λ ∈ N, where N is a projective variety whose irreducible components are copies of ℙ1 in one-to-one correspondence with the simple factors of G. If λ is the generic point of the ith irreducible component, [Formula: see text] is the universal subcategory corresponding to the ith simple factor of G. Matter, on the contrary, is encoded in the subcategories [Formula: see text] where {λa} is a finite set of closed points in N. In particular, for a Gaiotto theory there is one such family of subcategories, [Formula: see text], for each maximal degeneration of the corresponding surface Σ, and the index variety N may be identified with the degenerate Gaiotto surface itself: generic light subcategories correspond to cylinders, while closed-point subcategories to "fixtures" (spheres with three punctures of various kinds) and higher-order generalizations. The rules for "gluing" categories are more general than the geometric gluing of surfaces, allowing for a few additional exceptional [Formula: see text] theories which are not of the Gaiotto class. We include several examples and some amusing consequence, as the characterization in terms of quiver combinatorics of asymptotically free theories.

Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials

To each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential in such a way that whenever we apply a flip to a tagged triangulation the Jacobian algebra of the quiver with potential (QP) associated to the resulting tagged triangulation is isomorphic to the Jacobian algebra of the QP obtained by mutating the QP of the original one. Furthermore, we show that any two tagged triangulations are related by a sequence of flips compatible with QP-mutation. We also prove that, for each of the QPs constructed, the ideal of the non-completed path algebra generated by the cyclic derivatives is admissible and the corresponding quotient is isomorphic to the Jacobian algebra. These results, which generalize some of the second author’s previous work for ideal triangulations, are then applied to prove properties of cluster monomials, like linear independence, in the cluster algebra associated to the given surface by Fomin, Shapiro and Thurston (with an arbitrary system of coefficients).

Chebyshev curves, free resolutions and rational curve arrangements

First we construct a free resolution for the Milnor (or Jacobian) algebra M(f) of a complex projective Chebyshev plane curve d : f = 0 of degree d. In particular, this resolution implies that the dimensions of the graded components M(f)k are constant for k ≥ 2d − 3.Then we show that the Milnor algebra of a nodal plane curve C has such a behaviour if and only if all the irreducible components of C are rational.For the Chebyshev curves, all of these components are in addition smooth, hence they are lines or conics and explicit factorizations are given in this case.